Let F=Vf, where f(x.y.z)=x3y²z+z³, and let S be the surface boundary of the solid E={(x.y,z)ER³ : x?+y²+z?s1,x20, y20, z 20}. Let I= •n do. Let h(x.y)= -2x³+9x²-12x - y². Choose the TWO correct statements. (1,0) and (-1,0) are local extrema of h(x,y) h(x,y) has no local minima. (1,0) and (0,0) are local extrema of h(x,y) of h(x,y) has a saddle point at (0,0) h(x,y) has a local minimum at (2,0). 12

Q: Calculate F· dS where F = (3xy', ze", 2°) and S is the surface of the solid bounded by the cylinder…

A: We need to calculate ∬SF→.dS which is the flux. Now, we can use the divergence theorem to find…

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Q: Let S be the solid bounded by the planes z = o and y + z = 3, and the cylinder x2+y2 =2y-

A: answer is in next step

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A: Parametrize the three curves separately and c is the sum of the three integrals.

Q: The region W lies between the spheres x2+y2+z2=1 and x2+y2+z2=9 and within the cone z=x2+y2−−−−−−√…

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Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

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Q: 3. Evaluate if G is a solid in the first octant bounded by the plane y+ z = 2 and the surface y = 1–…

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Q: Evaluate the triple integral I/| (x, y, z) dV over the solid E. f(x, y, z) = vx2 + y², E = {(x, y,…

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Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

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A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

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A: By using the divergence theorem we solve the above question

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Q: Calculate F- dS where F = (3ry², re², z³) S is the surface of the solid bounded by the cylinder y? +…

A: The general form of a vector-valued function in three dimensions is given by F¯=F1,F2,F3, where F1,…

Q: (a) Given F(x, y, z) = –xi– y j+x²k and o is the surface of the solid bounded by the Vx² + y² and…

A: There is no specific introduction for this question. Please go through the solution.

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Q: please help me as soon as possbile

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Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

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Q: E) Sketch the solid bounded by the cylinder ** +y = 4 and the planes y+z = 4 and z = 0

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Q: Evaluate (x² + y) dv G if G is the solid bounded by the parabolic cylinder z = the planes x = 0, x =…

A: Here I am using simple tripple integration to solve this question.

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Q: Use Stoke's theorem to evaluate the integral xydx+x'dy – 4x’ydz where C is the boundary of the…

A: Here the surface is the plane of the square with vertices A1,0,1, B1,1,1, C0,1,1 and D0,0,1.  To…

Q: (a) Integrate g(x,y,z) = Inx²+y over the surface of the region cut from the plane 4x+3y+2z=1 by the…

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Q: Let S be the piece of the surface z = 9-r that lies inside the cylinder r2 +y² = 9, and let F = (x,…

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A: Use Gauss divergence theorem to compute the outward flux.

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Q: Consider the solid bounded by surfaces S1, S2, S3, S4, S5, and S6 with vertices A, B, C, D, E and O,…

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Q: (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 - x², y = 0, and y = 4

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